A few examples arising from the prior art and known solutions which are aimed at addressing this problem are considered below. A first illustration relates to the orbital control of a spacecraft, for example a telecommunication satellite of a few tonnes that one wishes to position precisely in the vicinity of a given point of the geostationary orbit, once on station. In the case of a chemically propelled satellite, orbital corrections must be made at regular time intervals (typically once a fortnight), in the form of speed increments (typically from 1 to 2 m/s) that have to be imparted to the satellite in each of the directions tangential to the orbit and perpendicular to the plane of the orbit. In the case considered here, these speed increments are carried out physically by virtue of the propulsion system mounted onboard the satellite, by expelling gas through thrusters, the total speed increment that is to be carried out being 50 m/s per year. The satellite control system calculates, according to a predefined algorithm, the opening orders to be dispatched to the gas ejection valves of the various thrusters so as to carry out the desired speed increment. A speed increment in a given direction may be carried out, according to the principle of action-reaction, by controlling the opening of the valve or valves of one or more thrusters ejecting gas in the direction opposite to the speed increment desired.
In the general case covering most space missions, the speed increments to be imparted to the satellites and space probes for orbital control typically range from a few cm/s to a few 100 m/s.
In the very great majority of space applications, the control for opening the valves is an on-off control and not a proportional control. Therefore, the thrust resulting from the opening of a valve is approximately constant (of the order of from 1 to 20 Newtons in the case of the orbital control of telecommunication satellites, from a few tenths to a few hundred Newtons in the general case). The control of the speed increment carried out is performed in open loop, on the basis of a thrust model of the thrusters, by controlling the total duration of opening of the valves of each active thruster. The simplest consists in controlling the opening of the valves in a continuous manner during the time just necessary for carrying out the entire desired speed increment, typically a few hundred seconds in the case considered. However, the continuous opening of one or more valves for such a duration is not recommended, since the perturbing torques resulting from the misalignments of the thrusts of the thrusters with respect to their nominal orientation could exceed the admissible limit and cause the satellite and its payloads to go off target by more than the limit permitted by the mission (typically 0.05 to 0.1 degrees for the application considered).
According to the prior art, in order to remedy this problem, the speed increment is carried out not in one go, in full, by continuously opening the thrusters concerned for the time span just necessary, but rather through a succession of small increments distributed over a larger time span, typically two to four times the duration just necessary in the case of a continuous thrust. The effect produced is on average equivalent to a reduction by the same factor of the equivalent thrust imparted to the satellite during the manoeuvre (correspondingly reducing the harmful transient effect of the perturbing torques), it is not possible for this to be produced through partial opening of the valves of the propulsion system which, let us recall, work according to our assumptions in on-off mode. In practice, according to the prior art, the control orders for opening the thrusters are sampled at a sampling period T. In each sampling interval lying between two successive sampling instants, each active thruster valve is controlled periodically at the period T to the open position in a time interval of duration ΔT that is strictly less than the sampling period T, and typically equal to an integer fraction of T. For reasons which will become apparent later, the valve opening order is preferably centred in the middle of the sampling interval. Thus, if it is conventionally considered that the control dispatched to each valve's opening system is equal to 1 to instruct total opening, and 0 to instruct total closure, the control signal dispatched to each valve of the propulsion system has the form of a periodic succession, of period T, of small increments of value 1 and of duration ΔT that are centred in the middle of the sampling intervals, the control being 0 outside of these durations ΔT (see below the example of FIG. 2a).
It shall be noted that the sampling period T should be chosen as large as possible (typically a few seconds) so as to minimize, during the lifetime of the satellite which is of the order of fifteen years, the total number of opening/closing transients of the valves of thrusters which are sensitive to this parameter. Additionally, this sampling period should be sufficiently large that the increments to be carried out are of a larger order of magnitude than the minimum duration of opening of the thrusters (which is called the Minimum Impulse Bit), below which there is a notable loss of effectiveness and a significant over consumption of fuel.
The advantage of a known method such as this is therefore to spread the achieving of the speed increment over a duration that is larger by a factor T/ΔT than the duration just necessary in the case of continuous opening of the thrusters, this having the mean effect of decreasing by the same factor the equivalent thrust imparted to the satellite, and hence of correspondingly reducing the undesirable transients due to the perturbing torques. However, against this decisive advantage that should be preserved, this method according to the prior art has a significant limitation, which may weaken or even cancel the anticipated benefit. This limitation is due to the fact that the valve opening control is periodic in nature on account of the spreading of the manoeuvre, as explained above. For example, for values T=1 second and ΔT=0.2 seconds, FIG. 4a (described below) presents the Fourier spectrum of the control profile according to the prior art presented above. This spectrum shows that the energy of the control is concentrated on the integer multiples k/T of the sampling frequency, giving rise to a risk of resonance between the thrust increments imparted to the satellite during the manoeuvre and the modes of vibration of flexible elements, such as, for example, the large solar generators, all the more so as the sampling frequency 1/T is relatively low, for the reasons explained above. It should be stressed that such resonance between an open-loop control for achieving speed increments and flexible modes of large appendages such as solar generators may be extremely harmful because these flexible modes are very weakly damped (damping factor of the order of 0.001).
According to the prior art, one seeks to offset this effect by selecting a sampling frequency 1/T such that its integer multiples are all far enough away from the frequencies of the flexible modes of the solar generators to avoid any harmful coincident or closeness. However, these flexible mode frequencies are known with a significant uncertainty of the order of from 10 to 20%, especially for high frequencies, and so in practice this does not make it possible to reduce the risk of coincidence. Moreover, the density of the flexible modes may be such that it is impossible to find a good location between these modes. Finally, it would be advantageous to increase the sampling frequency so as to push the coincidence problem out to higher frequencies, just where the energy of the flexible modes is not as significant and their effects are more limited, but this increase is constrained by the effects indicated above (number of opening/closing cycles of the thrusters, loss of effectiveness induced by overly short increments).
According to the prior art, it may also be sought to increase the bandwidth of the satellite attitude control so as to control at least the first modes of the flexible appendages and thus reduce the harmful effect of any resonances. However, this solution is also limited by several factors: level of control authority available (problem of saturation), risk of destabilization of the attitude control which must manage a large number of poorly known and very lightly damped flexible modes, prevailing risk of resonance with flexible modes situated outside of the bandwidth of the attitude control.
Another example of a conventional approach relates to the pointing of a large payload exhibiting flexible modes of fairly low frequency. Such is the case, for example, when large solar generators of telecommunication satellites such as those mentioned previously are rotating, having to execute one revolution in 24 hours with respect to the platform of the satellite so as to remain pointing towards the sun throughout the orbit, whilst the satellite remains pointing towards the Earth. These solar generators are set in rotation by stepper motors. The control of these motors is sampled with a period T. Every T seconds, an angular increment control is dispatched to the motor so as to achieve the desired rotation (as described hereinbelow with reference to FIG. 3a). The increments have nominally the same value. Here again, on account of the periodicity of the increments, there is a possible risk of resonance between the control and the flexible modes of the mechanical system to be controlled.